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Q-system Cluster Algebras, Paths and Total Positivity
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2010, том 6
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Q-system Cluster Algebras, Paths and Total Positivity
Інший ID:
2010 Mathematics Subject Classification: 05E10; 13F16; 82B20
Посилання:
Q-system Cluster Algebras, Paths and Total Positivity / P. di Francesco, R. Kedem // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 28 назв. — англ.
Підтримка:
This paper is a contribution to the Proceedings of the Workshop “Geometric Aspects of Discrete and UltraDiscrete Integrable Systems” (March 30 – April 3, 2009, University of Glasgow, UK). The full collection is available at http://www.emis.de/journals/SIGMA/GADUDIS2009.html.
We thank M. Gekhtman, S. Fomin, A. Postnikov, N. Reshetikhin and A. Vainshtein for useful discussions. RK’s research is funded in part by NSF grant DMS-0802511. RK thanks CEA/Saclay IPhT for their hospitality. PDF’s research is partly supported by the European network grant ENIGMA and the ANR grants GIMP and GranMa. PDF thanks the department of Mathematics of the University of Illinois at Urbana-Champaign for hospitality and support, and the department of Mathematics of the University of California Berkeley for hospitality
Дата:
2010
Переглядів:
766
Завантажень:
171
Анотація:
In the first part of this paper, we provide a concise review of our method of solution of the Ar Q-systems in terms of the partition function of paths on a weighted graph. In the second part, we show that it is possible to modify the graphs and transfer matrices so as to provide an explicit connection to the theory of planar networks introduced in the context of totally positive matrices by Fomin and Zelevinsky. As an illustration of the further generality of our method, we apply it to give a simple solution for the rank 2 affine cluster algebras studied by Caldero and Zelevinsky.
